Orthogonal Trajectories

Contents

Introduction

Exact Differential Equations

Equations reducible to exact form

Linear Differential Equations

Non-Linear Differential Equations

Applications

Newton’s law of cooling

Law of natural growth and decay

Orthogonal Trajectories

1M Questions

Two families of curves such that every member of either family cuts each member of the other family at right angles are called orthogonal trajectories of each other

Examples
  • The families of concentric circles with centres at the origin (\( x^2 + y^2 = c \)) and the family of straight lines passing through the origin (\( y = mx \)) are orthogonal trajectories to each other.
  • Meridians and parallels of the world globe.
  • Lines of heat flow and isothermal curves.
Self-Orthogonal Family of Curves

If each member of a given family of curves cuts every other member of the family at right angles, then the given family of curves is set to be self-orthogonal.

Working Rule to Find Orthogonal Trajectories in Cartesian Coordinates

Let \[ f(x, y, c) = 0 \quad …… (1) \] be the equation of given family of curves, where c is the parameter

  1. Form the differential equation \[ F(x, y, \frac{dy}{dx}) = 0 \quad ……(2) \] by eliminating the parameter \( c \) from (1).
  2. Replace \( \frac{dy}{dx} \) by \( -\frac{dx}{dy} \) in (2), then we get the differential equation of the family of orthogonal trajectories as

    3. \[ F\left(x, y, -\frac{dx}{dy}\right) = 0 \quad \text{…… (3)} \]

  3. Solve equation (3) to get the equation of the family of orthogonal trajectories of equation (1).

1. Find the Orthogonal Trajectories of the Family of Parabolas \( y^2 = 4ax \)

Solution: Given \[ y^2 = 4ax … (1) \] where \( a \) is the parameter.

Differentiating (1) with respect to \( x \), we get:

\[ 2y \frac{dy}{dx} = 4a \]

\[ \Rightarrow 2y \frac{dy}{dx} = \frac{y^2}{x} \quad \text{using equation (1)} \]

\[ \Rightarrow \frac{dy}{dx} = \frac{y}{2x} \quad … (2) \] which is the differential equation of the given family of curves.

Replace \( \frac{dy}{dx} \) by \( -\frac{dx}{dy} \) in (2), we get:

\[ -\frac{dx}{dy} = \frac{y}{2x} \]

\[ \Rightarrow y dy = -2x dx \quad\] which is the differential equation of orthogonal trajectories of the given family of curves.

Integrating the above equation, we get:

\[ \frac{y^2}{2} = -x^2 + c \]

or

\[ x^2 + \frac{y^2}{2} = c \]

Model Problems

Find the orthogonal trajectories or show the given properties for the following families of curves:

  1. Find the orthogonal trajectories of the family of semi-cubical parabolas \( y^2 = x^3 \).
  2. Find the orthogonal trajectories of the family of confocal conics \( \frac{x^2}{a^2 + \lambda} + \frac{y^2}{a^2 + \lambda} = 1 \), where \( \lambda \) is the parameter.
  3. Find the orthogonal trajectories of the family of circles \( x^2 + (y - c)^2 = r^2 \).
  4. Find the orthogonal trajectories of the family of curves \( x^{2/3} + y^{2/3} = a^{2/3} \).
  5. Show that \( \frac{x^2}{a^2 + \lambda} + \frac{y^2}{b^2 + \lambda} = 1 \) is self-orthogonal, where \( \lambda \) is the parameter.
  6. Show that the system of confocal and co-axial parabolas \( y^2 = 4a(x + a) \) is self-orthogonal.
  7. Show that the system of confocal and co-axial parabolas \( x^2 = 4a(y + a) \) is self-orthogonal.
  8. Show that the system of rectangular hyperbolas \( x^2 - y^2 = a^2 \) and \( xy = b^2 \) are mutually orthogonal trajectories.

Answers

  1. \( 2x^2 + 3y^2 = c \)
  2. \( x^2 + y^2 - 2a^2 \log x = c \)
  3. \( x^2 + y^2 = cx \)
  4. \( x^{4/3} - y^{4/3} = a^{4/3} \)